I'm amazed at how many people in this thread answer the question with such certainty that they're right, and that the opposition consists of imbeciles who don't know what they're talking about.
It's like the Laver vs. Borg vs. Sampras vs. Federer vs. Nadal arguments. They're the only five with a legitimate claim in tennis, and each has his proponents. I've heard, over the years, with various quality arguments in support of each, Morphy, Lasker, Capablanca, Alekhine, Tal, Fischer, Karpov, Kasparov and Carlsen. (I've even heard that if playing with no time constraints Reshevsky was probably the greatest of all time.)
Until such time as every player has access to equivalent resources and opportunities, we have no idea who would win with each at the height of their powers.
Frankly, I think Fischer would beat Carlsen ... but that opinion isn't even worth the time it took to type it, because someone else thinking the opposite has just as valid an opinion—that is to say, utterly meaningless.
Don't mind me, just testing my first post here.